Ex 10.2, 15 - Chapter 10 Class 12 Vector Algebra

Transcript

Ex 10.2, 15 Find the position vector of a point R which divides the line joining two points P and Q whose position vectors are 𝑖 ̂ + 2𝑗 ̂ − 𝑘 ̂ and – 𝑖 ̂ + 𝑗 ̂ + 𝑘 ̂ respectively, in the ratio 2 : 1 (i) internally (𝑂𝑃) ⃗= 𝑖 ̂ + 2𝑗 ̂ − 𝑘 ̂ (𝑂𝑄) ⃗= – 𝑖 ̂ + 𝑗 ̂ + 𝑘 ̂ Position vector of R = (2.(𝑂𝑄) ⃗ + 1.(𝑂𝑃) ⃗)/(2 + 1) (𝑂𝑅) ⃗ = (2(−𝑖 ̂" + " 𝑗 ̂" + " 𝑘 ̂ ) + 1(𝑖 ̂" +" 2𝑗 ̂" + " 𝑘 ̂))/3 = (−2𝑖 ̂ + 2𝑗 ̂ + 2𝑘 ̂ + 𝑖 ̂ + 2𝑗 ̂ −( 𝑘) ̂)/3 = (−𝑖 ̂ + 4𝑗 ̂ + 𝑘 ̂)/3 Thus, position vector of R dividing P and Q internally is (−𝟏)/𝟑 𝒊 ̂ + 𝟒/𝟑 𝒊 ̂ + 𝟏/𝟑 𝒌 ̂ Ex 10.2, 15 Find the position vector of a point R which divides the line joining two points P and Q whose position vectors are 𝑖 ̂ + 2 𝑗 ̂ − 𝑘 ̂ and – 𝑖 ̂ + 𝑗 ̂ + 𝑘 ̂ respectively, in the ratio 2 : 1 (ii) externally Position vector of R = (2(𝑂𝑄) ⃗ − 1(𝑂𝑃) ⃗)/(2 − 1) (𝑂𝑅) ⃗ = (2(−𝑖 ̂" + " 𝑗 ̂" + " 𝑘 ̂ )−1(𝑖 ̂" + " 2𝑗 ̂" − " 𝑘 ̂))/(2−1) (𝑂𝑅) ⃗ = (−2𝑖 ̂ + 2𝑗 ̂ + 2𝑘 ̂ − 𝑖 ̂ − 2𝑗 ̂ + 𝑘 ̂)/(2−1) (𝑂𝑅) ⃗ = –3𝒊 ̂ + 3𝒌 ̂ Thus, Position vector of R dividing P and Q externally is −3𝑖 ̂ + 3𝑘 ̂